5Differentiability
IA Analysis I
5.4 Complex differentiation
Definition (Complex differentiability). Let
f
:
C → C
. Then
f
is differentiable
at z with derivative f
0
(z) if
lim
h→0
f(z + h) − f (z)
h
exists and equals f
0
(z).
Equivalently,
f(z + h) = f(z) + hf
0
(z) + o(h).
This is exactly the same definition with real differentiation, but has very
different properties!
All the usual rules — chain rule, product rule etc. also apply (with the same
proofs). Also the derivatives of polynomials are what you expect. However, there
are some more interesting cases.
Example. f(z) = ¯z is not differentiable.
z + h − z
h
=
¯
h
h
=
(
1 h is real
−1 h is purely imaginary
If this seems weird, this is because we often think of
C
as
R
2
, but they are
not the same. For example, reflection is a linear map in
R
2
, but not in
C
. A
linear map in
C
is something in the form
x 7→ bx
, which can only be a dilation
or rotation, not reflections or other weird things.
Example.
f
(
z
) =
|z|
is also not differentiable. If it were, then
|z|
2
would be as
well (by the product rule). So would
|z|
2
z
=
¯z
when
z 6
= 0 by the quotient rule.
At
z
= 0, it is certainly not differentiable, since it is not even differentiable on
R
.